Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geomatics Engineering Or Surveying

Engineering Mechanics

Hydrology

Transportation Engineering

Strength of Materials Or Solid Mechanics

Reinforced Cement Concrete

Steel Structures

Irrigation

Environmental Engineering

Engineering Mathematics

Structural Analysis

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

General Aptitude

1

Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements
is true ?

A

The lines are not concurrent

B

The lines are concurrent at the point $$\left( {{3 \over 4},{1 \over 2}} \right)$$

C

The lines are all parallel

D

Each line passes through the origin

Equation of lines;

px + qy + r = 0 . . . . . (1)

Also given

3p + 2q + 4r = 0 . . . . . . (2)

divide equation (2) by 4, we get

$${3 \over 4}P + {2 \over 4}q + r = 0$$ . . . . (3)

By comparing (1) and (3) we get,

x = $${3 \over 4}$$ and y = $${2 \over 4}$$ = $${1 \over 2}$$

For any value of p,q and r, the equation of set of lines will pan through $$\left( {{3 \over 4},{1 \over 2}} \right)$$

px + qy + r = 0 . . . . . (1)

Also given

3p + 2q + 4r = 0 . . . . . . (2)

divide equation (2) by 4, we get

$${3 \over 4}P + {2 \over 4}q + r = 0$$ . . . . (3)

By comparing (1) and (3) we get,

x = $${3 \over 4}$$ and y = $${2 \over 4}$$ = $${1 \over 2}$$

For any value of p,q and r, the equation of set of lines will pan through $$\left( {{3 \over 4},{1 \over 2}} \right)$$

2

Let the equations of two sides of a triangle be 3x $$-$$ 2y + 6 = 0 and 4x + 5y $$-$$ 20 = 0. If the orthocentre of this triangle is at (1, 1), then the equation of its third side is :

A

122y $$-$$ 26x $$-$$ 1675 = 0

B

122y + 26x + 1675 = 0

C

26x + 61y + 1675 = 0

D

26x $$-$$ 122y $$-$$ 1675 = 0

4x + 5y $$-$$ 20 = 0 . . .(1)

3x $$-$$ 2y + 6 = 0 . . . (2)

orthocentre is (1, 1)

line perpendicular to 4x + 5y $$-$$ 20 = 0

and passes through (1, 1) is

(y $$-$$ 1) = $${5 \over 4}$$(x $$-$$ 1)

$$ \Rightarrow $$ 5x $$-$$ 4y = 1 . . .(3)

and line $$ \bot $$ to 3x $$-$$ 2y + 6 = 0

and passes through (1, 1)

y $$-$$ 1 = $$-$$ $${2 \over 3}$$ (x $$-$$ 1)

$$ \Rightarrow $$ 2x + 3y = 5 . . .(4)

Solving (1) and (4) we get C$$\left( {{{35} \over 2}, - 10} \right)$$

Solving (2) and (3) we get A $$\left( { - 13,{{ - 33} \over 2}} \right)$$

Side BC is y + 10 = $${{{{ - 33} \over 2} + 10} \over { - 13 - {{35} \over 2}}}\left( {x - {{35} \over 2}} \right)$$

$$ \Rightarrow $$ y + 10 = $${{13} \over {61}}\left( {x - {{35} \over 2}} \right)$$

$$ \Rightarrow $$ 26x $$-$$ 122y $$-$$ 1675 = 0

3

If 5, 5r, 5r^{2} are the lengths of the sides of a triangle, then r cannot be equal to -

A

$${7 \over 4}$$

B

$${5 \over 4}$$

C

$${3 \over 4}$$

D

$${3 \over 2}$$

r = 1 is obviously true.

Let 0 < r < 1

$$ \Rightarrow $$ r + r^{2} > 1

$$ \Rightarrow $$ r^{2} + r $$-$$ 1 > 0

$$\left( {r - {{ - 1 - \sqrt 5 } \over 2}} \right)\left( {r - \left( {{{ - 1 + \sqrt 5 } \over 2}} \right)} \right)$$

$$ \Rightarrow r - {{ - 1 - \sqrt 5 } \over 2}$$ or $$r > {{ - 1 + \sqrt 5 } \over 2}$$

$$r \in \left( {{{\sqrt 5 - 1} \over 2},1} \right)$$

$${{\sqrt 5 - 1} \over 2} < r < 1$$

When r > 1

$$ \Rightarrow {{\sqrt 5 + 1} \over 2} > {1 \over r} > 1$$

$$ \Rightarrow r \in \left( {{{\sqrt 5 - 1} \over 2},{{\sqrt 5 + 1} \over 2}} \right)$$

Now check options

Let 0 < r < 1

$$ \Rightarrow $$ r + r

$$ \Rightarrow $$ r

$$\left( {r - {{ - 1 - \sqrt 5 } \over 2}} \right)\left( {r - \left( {{{ - 1 + \sqrt 5 } \over 2}} \right)} \right)$$

$$ \Rightarrow r - {{ - 1 - \sqrt 5 } \over 2}$$ or $$r > {{ - 1 + \sqrt 5 } \over 2}$$

$$r \in \left( {{{\sqrt 5 - 1} \over 2},1} \right)$$

$${{\sqrt 5 - 1} \over 2} < r < 1$$

When r > 1

$$ \Rightarrow {{\sqrt 5 + 1} \over 2} > {1 \over r} > 1$$

$$ \Rightarrow r \in \left( {{{\sqrt 5 - 1} \over 2},{{\sqrt 5 + 1} \over 2}} \right)$$

Now check options

4

A point P moves on the line 2x – 3y + 4 = 0. If Q(1, 4) and R (3, – 2) are fixed points, then the locus of the centroid of $$\Delta $$PQR is a line -

A

parallel to y-axis

B

with slope $${2 \over 3}$$

C

parallel to x-axis

D

with slope $${3 \over 2}$$

Let the centroid of $$\Delta $$PQR is (h, k) & P is ($$\alpha $$, $$\beta $$), then

$${{\alpha + 1 + 3} \over 3} = h\,$$ and $${{\beta + 4 - 2} \over 3} = k$$

$$\alpha = \left( {3h - 4} \right)$$ $$\beta = \left( {3k - 4} \right)$$

Point P($$\alpha $$, $$\beta $$) lies on the line 2x $$-$$ 3y + 4 = 0

$$ \therefore $$ 2(3h $$-$$ 4) $$-$$ 3 (3k $$-$$ 2) + 4 = 0

$$ \Rightarrow $$ locus is 6x $$-$$ 9y + 2 = 0

$${{\alpha + 1 + 3} \over 3} = h\,$$ and $${{\beta + 4 - 2} \over 3} = k$$

$$\alpha = \left( {3h - 4} \right)$$ $$\beta = \left( {3k - 4} \right)$$

Point P($$\alpha $$, $$\beta $$) lies on the line 2x $$-$$ 3y + 4 = 0

$$ \therefore $$ 2(3h $$-$$ 4) $$-$$ 3 (3k $$-$$ 2) + 4 = 0

$$ \Rightarrow $$ locus is 6x $$-$$ 9y + 2 = 0

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (4) *keyboard_arrow_right*

AIEEE 2003 (5) *keyboard_arrow_right*

AIEEE 2004 (4) *keyboard_arrow_right*

AIEEE 2005 (2) *keyboard_arrow_right*

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JEE Main 2013 (Offline) (2) *keyboard_arrow_right*

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JEE Main 2021 (Online) 24th February Morning Slot (1) *keyboard_arrow_right*

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Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *keyboard_arrow_right*

Complex Numbers *keyboard_arrow_right*

Quadratic Equation and Inequalities *keyboard_arrow_right*

Permutations and Combinations *keyboard_arrow_right*

Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

Sequences and Series *keyboard_arrow_right*

Matrices and Determinants *keyboard_arrow_right*

Vector Algebra and 3D Geometry *keyboard_arrow_right*

Probability *keyboard_arrow_right*

Statistics *keyboard_arrow_right*

Mathematical Reasoning *keyboard_arrow_right*

Functions *keyboard_arrow_right*

Limits, Continuity and Differentiability *keyboard_arrow_right*

Differentiation *keyboard_arrow_right*

Application of Derivatives *keyboard_arrow_right*

Indefinite Integrals *keyboard_arrow_right*

Definite Integrals and Applications of Integrals *keyboard_arrow_right*

Differential Equations *keyboard_arrow_right*

Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*